MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe1tmmul2 Structured version   Visualization version   GIF version

Theorem coe1tmmul2 21545
Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z 0 = (0g𝑅)
coe1tm.k 𝐾 = (Base‘𝑅)
coe1tm.p 𝑃 = (Poly1𝑅)
coe1tm.x 𝑋 = (var1𝑅)
coe1tm.m · = ( ·𝑠𝑃)
coe1tm.n 𝑁 = (mulGrp‘𝑃)
coe1tm.e = (.g𝑁)
coe1tmmul.b 𝐵 = (Base‘𝑃)
coe1tmmul.t = (.r𝑃)
coe1tmmul.u × = (.r𝑅)
coe1tmmul.a (𝜑𝐴𝐵)
coe1tmmul.r (𝜑𝑅 ∈ Ring)
coe1tmmul.c (𝜑𝐶𝐾)
coe1tmmul.d (𝜑𝐷 ∈ ℕ0)
Assertion
Ref Expression
coe1tmmul2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥,   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥,
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 coe1tmmul.r . . 3 (𝜑𝑅 ∈ Ring)
2 coe1tmmul.a . . 3 (𝜑𝐴𝐵)
3 coe1tmmul.c . . . 4 (𝜑𝐶𝐾)
4 coe1tmmul.d . . . 4 (𝜑𝐷 ∈ ℕ0)
5 coe1tm.k . . . . 5 𝐾 = (Base‘𝑅)
6 coe1tm.p . . . . 5 𝑃 = (Poly1𝑅)
7 coe1tm.x . . . . 5 𝑋 = (var1𝑅)
8 coe1tm.m . . . . 5 · = ( ·𝑠𝑃)
9 coe1tm.n . . . . 5 𝑁 = (mulGrp‘𝑃)
10 coe1tm.e . . . . 5 = (.g𝑁)
11 coe1tmmul.b . . . . 5 𝐵 = (Base‘𝑃)
125, 6, 7, 8, 9, 10, 11ply1tmcl 21541 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
131, 3, 4, 12syl3anc 1370 . . 3 (𝜑 → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
14 coe1tmmul.t . . . 4 = (.r𝑃)
15 coe1tmmul.u . . . 4 × = (.r𝑅)
166, 14, 15, 11coe1mul 21539 . . 3 ((𝑅 ∈ Ring ∧ 𝐴𝐵 ∧ (𝐶 · (𝐷 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
171, 2, 13, 16syl3anc 1370 . 2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
18 eqeq2 2748 . . . 4 ((((coe1𝐴)‘(𝑥𝐷)) × 𝐶) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
19 eqeq2 2748 . . . 4 ( 0 = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
20 coe1tm.z . . . . . . 7 0 = (0g𝑅)
211adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Ring)
22 ringmnd 19880 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Mnd)
24 ovex 7362 . . . . . . . 8 (0...𝑥) ∈ V
2524a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0...𝑥) ∈ V)
26 simprr 770 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷𝑥)
274adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℕ0)
28 simprl 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℕ0)
29 nn0sub 12376 . . . . . . . . . 10 ((𝐷 ∈ ℕ0𝑥 ∈ ℕ0) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3027, 28, 29syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3126, 30mpbid 231 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ ℕ0)
3227nn0ge0d 12389 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 0 ≤ 𝐷)
33 nn0re 12335 . . . . . . . . . . 11 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
3433ad2antrl 725 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℝ)
354nn0red 12387 . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
3635adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℝ)
3734, 36subge02d 11660 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0 ≤ 𝐷 ↔ (𝑥𝐷) ≤ 𝑥))
3832, 37mpbid 231 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ≤ 𝑥)
39 fznn0 13441 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4039ad2antrl 725 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4131, 38, 40mpbir2and 710 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ (0...𝑥))
421ad2antrr 723 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43 eqid 2736 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
4443, 11, 6, 5coe1f 21480 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
452, 44syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4645ad2antrr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1𝐴):ℕ0𝐾)
47 elfznn0 13442 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
4847adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
4946, 48ffvelcdmd 7012 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
50 eqid 2736 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
5150, 11, 6, 5coe1f 21480 . . . . . . . . . . . 12 ((𝐶 · (𝐷 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5213, 51syl 17 . . . . . . . . . . 11 (𝜑 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5352ad2antrr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
54 fznn0sub 13381 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
5554adantl 482 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥𝑦) ∈ ℕ0)
5653, 55ffvelcdmd 7012 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾)
575, 15ringcl 19887 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5842, 49, 56, 57syl3anc 1370 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5958fmpttd 7039 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
601ad2antrr 723 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝑅 ∈ Ring)
613ad2antrr 723 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐶𝐾)
624ad2antrr 723 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ∈ ℕ0)
63 eldifi 4072 . . . . . . . . . . . . 13 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → 𝑦 ∈ (0...𝑥))
6463, 54syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → (𝑥𝑦) ∈ ℕ0)
6564adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (𝑥𝑦) ∈ ℕ0)
66 eldifsn 4733 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷)))
67 simplrl 774 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
6867nn0cnd 12388 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
6947nn0cnd 12388 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
7069adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
7168, 70nncand 11430 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥𝑦)) = 𝑦)
7271eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥𝑦)))
73 oveq2 7337 . . . . . . . . . . . . . . . 16 (𝐷 = (𝑥𝑦) → (𝑥𝐷) = (𝑥 − (𝑥𝑦)))
7473eqeq2d 2747 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥𝑦) → (𝑦 = (𝑥𝐷) ↔ 𝑦 = (𝑥 − (𝑥𝑦))))
7572, 74syl5ibrcom 246 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥𝑦) → 𝑦 = (𝑥𝐷)))
7675necon3d 2961 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥𝐷) → 𝐷 ≠ (𝑥𝑦)))
7776impr 455 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷))) → 𝐷 ≠ (𝑥𝑦))
7866, 77sylan2b 594 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ≠ (𝑥𝑦))
7920, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78coe1tmfv2 21544 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
8079oveq2d 7345 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
815, 15, 20ringrz 19914 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8242, 49, 81syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8363, 82sylan2 593 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8480, 83eqtrd 2776 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
8584, 25suppss2 8078 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) supp 0 ) ⊆ {(𝑥𝐷)})
865, 20, 23, 25, 41, 59, 85gsumpt 19650 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)))
87 fveq2 6819 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1𝐴)‘𝑦) = ((coe1𝐴)‘(𝑥𝐷)))
88 oveq2 7337 . . . . . . . . . 10 (𝑦 = (𝑥𝐷) → (𝑥𝑦) = (𝑥 − (𝑥𝐷)))
8988fveq2d 6823 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))))
9087, 89oveq12d 7347 . . . . . . . 8 (𝑦 = (𝑥𝐷) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
91 eqid 2736 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))
92 ovex 7362 . . . . . . . 8 (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) ∈ V
9390, 91, 92fvmpt 6925 . . . . . . 7 ((𝑥𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9441, 93syl 17 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9528nn0cnd 12388 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℂ)
9627nn0cnd 12388 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℂ)
9795, 96nncand 11430 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥 − (𝑥𝐷)) = 𝐷)
9897fveq2d 6823 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
993adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐶𝐾)
10020, 5, 6, 7, 8, 9, 10coe1tmfv1 21543 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10121, 99, 27, 100syl3anc 1370 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10298, 101eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = 𝐶)
103102oveq2d 7345 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
10486, 94, 1033eqtrd 2780 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
105104anassrs 468 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
1061ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
1073ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐶𝐾)
1084ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
10954ad2antll 726 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℕ0)
11054nn0red 12387 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℝ)
111110ad2antll 726 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℝ)
11233ad2antlr 724 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
11335ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
11447ad2antll 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115114nn0ge0d 12389 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
11647nn0red 12387 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117116ad2antll 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118112, 117subge02d 11660 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥𝑦) ≤ 𝑥))
119115, 118mpbid 231 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ≤ 𝑥)
120 simprl 768 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ¬ 𝐷𝑥)
121112, 113ltnled 11215 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷𝑥))
122120, 121mpbird 256 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123111, 112, 113, 119, 122lelttrd 11226 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) < 𝐷)
124111, 123gtned 11203 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥𝑦))
12520, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124coe1tmfv2 21544 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
126125oveq2d 7345 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
12745ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (coe1𝐴):ℕ0𝐾)
128127, 114ffvelcdmd 7012 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
129106, 128, 81syl2anc 584 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
130126, 129eqtrd 2776 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
131130anassrs 468 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
132131mpteq2dva 5189 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133132oveq2d 7345 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
1341, 22syl 17 . . . . . . 7 (𝜑𝑅 ∈ Mnd)
13520gsumz 18563 . . . . . . 7 ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136134, 24, 135sylancl 586 . . . . . 6 (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137136ad2antrr 723 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138133, 137eqtrd 2776 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 )
13918, 19, 105, 138ifbothda 4510 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ))
140139mpteq2dva 5189 . 2 (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
14117, 140eqtrd 2776 1 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wne 2940  Vcvv 3441  cdif 3894  ifcif 4472  {csn 4572   class class class wbr 5089  cmpt 5172  wf 6469  cfv 6473  (class class class)co 7329  cc 10962  cr 10963  0cc0 10964   < clt 11102  cle 11103  cmin 11298  0cn0 12326  ...cfz 13332  Basecbs 17001  .rcmulr 17052   ·𝑠 cvsca 17055  0gc0g 17239   Σg cgsu 17240  Mndcmnd 18474  .gcmg 18788  mulGrpcmgp 19807  Ringcrg 19870  var1cv1 21445  Poly1cpl1 21446  coe1cco1 21447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-of 7587  df-ofr 7588  df-om 7773  df-1st 7891  df-2nd 7892  df-supp 8040  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-map 8680  df-pm 8681  df-ixp 8749  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-fsupp 9219  df-oi 9359  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-7 12134  df-8 12135  df-9 12136  df-n0 12327  df-z 12413  df-dec 12531  df-uz 12676  df-fz 13333  df-fzo 13476  df-seq 13815  df-hash 14138  df-struct 16937  df-sets 16954  df-slot 16972  df-ndx 16984  df-base 17002  df-ress 17031  df-plusg 17064  df-mulr 17065  df-sca 17067  df-vsca 17068  df-tset 17070  df-ple 17071  df-0g 17241  df-gsum 17242  df-mre 17384  df-mrc 17385  df-acs 17387  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-mhm 18519  df-submnd 18520  df-grp 18668  df-minusg 18669  df-sbg 18670  df-mulg 18789  df-subg 18840  df-ghm 18920  df-cntz 19011  df-cmn 19475  df-abl 19476  df-mgp 19808  df-ur 19825  df-ring 19872  df-subrg 20119  df-lmod 20223  df-lss 20292  df-psr 21210  df-mvr 21211  df-mpl 21212  df-opsr 21214  df-psr1 21449  df-vr1 21450  df-ply1 21451  df-coe1 21452
This theorem is referenced by:  coe1tmmul2fv  21547  coe1sclmul2  21553
  Copyright terms: Public domain W3C validator